Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems

Abstract

The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.

Three nontrivial solutions for nonlinear fractional Laplacian equations

“…We recall here some well-known results which will be our major tools, mainly following [28] (see also [32]). Let (X, · ) be a reflexive Banach space, (X * , · * ) be its topological dual, and ϕ ∈ C 1 (X) be a functional.…”

“…where H k (·, ·) is the k-th (singular) homology group of a topological pair (see [28,Definition 6.9]). All these groups are real linear spaces.…”

“…Otherwise, we could find a sequence (u n ) in H s 0 (Ω) s.t. u n − u + = r for all n ∈ N, ϕ(u n ) → ϕ(u + ) and ϕ ′ (u n ) → 0 in H −s (Ω) (see [28,Corollary 5.12]). Then, by (C) we would have u n →ū in H s 0 (Ω) for someū ∈ H s 0 (Ω), ū − u + = r, hence in turn ϕ(ū) = ϕ(u + ), a contradiction.…”

“…Now we apply the implicit function theorem [28,Theorem 7.3] to the function (µ, u) → ϕ(µu) defined in (1, ∞) × S. By (4.9) we have for all (µ, u) ∈ (1, ∞) × S with ϕ(µu) = c…”

“…, and the latter is 0 by [28,Propositions 6.24,6.25] (recall that S is contractible in itself, as dim H s 0 (Ω) = ∞). Thus we have (4.7).…”